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arXiv:cond-mat/0005354 · Europhysics Letters (EPL)

Spin and charge Tomonaga-Luttinger parameters in quantum wires

C. E. Creffield, Wolfgang Häusler, Allan H. MacDonald

Published Jan 1, 2001• 25 citations
arXivHTMLPDF

Abstract

Low-energy properties of the homogeneous electron gas in one dimension are completely described by the group velocities of its charge (plasmon) and spin collective excitations. Because of the long range of the electron-electron interaction, the plasmon velocity is dominated by an electrostatic contribution and can be estimated accurately. We report on Quantum Monte Carlo simulations which demonstrate that the spin velocity is substantially decreased by interactions in semiconductor quantum wire realizations of the one-dimensional electron liquid.

DOI
10.1209/epl/i2001-00140-7

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Low-energy properties of the homogeneous electron gas in one dimension are completely described by the group velocities of its charge (plasmon) and spin collective excitations. Because of the long range of the electron-electron interaction, the plasmon velocity is dominated by an electrostatic contribution and can be estimated accurately. We report on Quantum Monte Carlo simulations which demonstrate that the spin velocity is substantially decreased by interactions in semiconductor quantum wire realizations of the one-dimensional electron liquid.
The low-energy properties of a homogeneous one-dimensional electron system are completely specified by two Tomonaga-Luttinger parameters ${K}_{\ensuremath{\rho}}$ and ${v}_{\ensuremath{\sigma}}.$ In this paper we discuss microscopic estimates of the values of these parameters in semiconductor quantum wires that exploit their relationship to thermodynamic properties. Motivated by the recognized similarity between correlations in the ground state of a one-dimensional electron liquid and correlations in a Wigner crystal, we evaluate these thermodynamic quantities in a self-consistent Hartree-Fock approximation. According to our calculations, the Hartree-Fock approximation ground state is a Wigner crystal at all electron densities and has antiferromagnetic order that gradually evolves from spin-density wave to localized in character as the density is lowered. Our results for ${K}_{\ensuremath{\rho}}$ are in good agreement with weak-coupling perturbative estimates ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}$ at high densities, but deviate strongly at low densities, especially when the electron-electron interaction is screened at long distances. ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}\ensuremath{\sim}{n}^{1/2}$ vanishes at small carrier density n, whereas we conjecture that ${K}_{\ensuremath{\rho}}\ensuremath{\rightarrow}1/2$ when $\stackrel{\ensuremath{\rightarrow}}{n}0,$ implying that ${K}_{\ensuremath{\rho}}$ should pass through a minimum at an intermediate density. Observation of this nonmonotonic dependence could be used to measure the effective interaction range in a realistic semiconductor quantum wire geometry. In the spin sector we find that the spin velocity decreases with increasing interaction strength or decreasing n. Strong correlation effects make it difficult to obtain fully consistent estimates of ${v}_{\ensuremath{\sigma}}$ from Hartree-Fock calculations. We conjecture that ${v}_{\ensuremath{\sigma}}{/v}_{\mathrm{F}}\ensuremath{\propto}{n/V}_{0},$ where V0{V}_{0}V0​ is the interaction strength, in the limit $\stackrel{\ensuremath{\rightarrow}}{n}0.$
In a one-dimensional (1D) system of interacting electrons, excitations of spin and charge travel at different speeds, according to the theory of a Tomonaga-Luttinger liquid (TLL) at low energies. However, the clear observation of this spin-charge separation is an ongoing challenge experimentally. We have fabricated an electrostatically gated 1D system in which we observe spin-charge separation and also the predicted power-law suppression of tunneling into the 1D system. The spin-charge separation persists even beyond the low-energy regime where the TLL approximation should hold. TLL effects should therefore also be important in similar, but shorter, electrostatically gated wires, where interaction effects are being studied extensively worldwide.
In this work we discuss extensions of the pioneering analysis by Dzyaloshinski ǐ and Larkin [Sov.Phys.JETP 38, 202 (1974)] of correlation functions for one-dimensional Fermi systems, focusing on the effects of quasiparticle relaxation enabled by a nonlinear dispersion.Throughout the work we employ both, the weakly interacting Fermi gas picture and nonlinear Luttinger liquid theory to describe attenuation of excitations and explore the fermion-boson duality between both approaches.Special attention is devoted to the role of spin-exchange processes, effects of interaction screening, and integrability.Thermalization rates for electron-and hole-like quasiparticles, as well as the decay rate of collective plasmon excitations and the momentum space mobility of spin excitations are calculated for various temperature regimes.The phenomenon of spin-charge drag is considered and the corresponding momentum transfer rate is determined.We further discuss how momentum relaxation due to several competing mechanisms, viz.triple electron collisions, electron-phonon scattering, and long-range inhomogeneities affect transport properties, and highlight energy transfer facilitated by plasmons from the perspective of the inhomogeneous Luttinger liquid model.Finally, we derive the full matrix of thermoelectric coefficients at the quantum critical point of the first conductance plateau transition, and address magnetoconductance in ballistic semiconductor nanowires with strong Rashba spin-orbit coupling.
Experiments on quasi-one-dimensional systems such as quantum wires and metallic chains on surfaces suggest the existence of electron-electron interactions of substantial range and hence physics beyond the Hubbard model. We therefore investigate one-dimensional, quarter-filled chains with a Coulomb potential with variable screening length by quantum Monte Carlo methods and exact diagonalization. The Luttinger liquid interaction parameter ${K}_{\ensuremath{\rho}}$ decreases with increasing interaction strength and range. Experimentally observed values close to 1/4 require strong interactions and/or large screening lengths. As predicted by bosonization, we find a metal-insulator transition at ${K}_{\ensuremath{\rho}}=1/4$. Upon increasing the screening length, the charge and spin correlation functions reveal the crossover from dominant 2kF2{k}_{\mathrm{F}}2kF​ spin correlations to dominant 4kF4{k}_{\mathrm{F}}4kF​ charge correlations, and a strong enhancement of the charge velocity. In the metallic phase, the signatures of spin-charge separation in the single-particle spectrum, spinon and holon bands, remain robust even for rather long-ranged interactions. The charge-density-wave state exhibits backfolded shadow bands.
Ultracold atoms confined to periodic potentials have proven to be a powerful tool for quantum simulation of complex many-body systems. We confine fermions to one dimension to realize the Tomonaga-Luttinger liquid model, which describes the highly collective nature of their low-energy excitations. We use Bragg spectroscopy to directly excite either the spin or charge waves for various strengths of repulsive interaction. We observe that the velocity of the spin and charge excitations shift in opposite directions with increasing interaction, a hallmark of spin-charge separation. The excitation spectra are in quantitative agreement with the exact solution of the Yang-Gaudin model and the Tomonaga-Luttinger liquid theory. Furthermore, we identify effects of nonlinear corrections to this theory that arise from band curvature and back-scattering.
In contrast to the well-known Fermi-liquid theory of three dimensions, interacting one-dimensional and quasi-one-dimensional systems of fermions are described at low energy by an effective theory known as Luttinger liquid theory. This theory is expressed in terms of collective many-body excitations that show exotic behavior such as spin-charge separation. Luttinger liquid theory is commonly applied on the premise that ``low energy'' describes both the spin and charge sectors. However, when the interactions in the system are very strong, as they typically are at low particle densities, the ratio of spin to charge energy may become exponentially small. It is then possible at very low temperatures for the single-spin excitation energy to be low compared to the characteristic single excitation charge energy, but still high compared to the characteristic spin energy. This energy window of near ground-state charge degrees of freedom but highly thermally excited spin degrees of freedom is called a spin-incoherent Luttinger liquid. The spin-incoherent Luttinger liquid exhibits a higher degree of universality than the Luttinger liquid and its properties are qualitatively distinct. In this Colloquium some recent theoretical developments in the field are detailed and experimental indications of such a regime in gated semiconductor quantum wires are described.